G. D. Plotkin. Lambda-definability in the full type hierarchy. In To H.B.Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, 1980.  Home/Search   Document Not in Database   Summary   Related Articles   Check

.... glueing or Freyd covers, see Lambek and Scott [LS86] To our knowledge, the first application of this method to type disciplines is given in Appendix C of Lafont [Laf88] In the case of simple types, this method corresponds closely to so called logical relations, described for instance in Plotkin [Plo80], Statman [Sta85] and Mitchell [Mit90] This correspondence is examined in detail. In the case of polymorphic types, a central role is played by relators, i.e. maps that take objects to objects and relations to relations. Sconing may be used to express relators as functors. jcm cs.stanford.edu ....

....4.4. In the general setting that would yield the above diagram with A Theta B instead of C and with D instead of C. In this section we compare this categorical generalization with three forms of logical relations that have already appeared in the literature, Kripke logical relations [Plo80, MM91], cpo logical relations [MS76, Rey74, CP92] and the relational setting over PER models discussed in Section 4 of [BFS90] Kripke logical relations over ordinary Henkin models were first used in [Plo80] in a characterization of lambda definability. Kripke logical relations were then adapted to ....

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G.D. Plotkin. Lambda definability in the full type hierarchy. In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 363--373. Academic Press, 1980.

.... and that this uniformity may be captured by imposing a relational structure on the types [OHT93, MSd93, MaR91, Wad89, Rey83, Str67] The other line of research concerns lambda definability and the full abstraction problem for various models of languages based on simply typed lambda calculus [JuT93, MaR91, MSd93, OHR94, Pit93, Plo80, Lau70]. Early attempts to characterize lambda definability in the full type hiearchy focused on invariance properties of functions that are definable by lambda terms. Invariance under permutation [Lau70] was the most obvious of these since lambda definable functions cannot speak about particular ....

....focused on invariance properties of functions that are definable by lambda terms. Invariance under permutation [Lau70] was the most obvious of these since lambda definable functions cannot speak about particular elements, but this was not enough for a complete characterization. A later attempt in [Plo80] introduced the idea of invariance under a logical relation and then, in the same paper, the notion of invariance under I relation by which Plotkin succeeded in characterizing lambda definability in certain full type hierarchies. More recently, Jung and Tiuryn describe a notion of logical ....

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G. D. Plotkin. Lambda-definability in the full type hierarchy. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, pages 363-373. Academic Press, 1980.

....4 contains a rather more detailed discussion of how correctness allows us to reason in a useful way about the standard interpretation of terms. The formal proof of correctness of B uses the following theorem, due in this particular form to [Abr90] Proposition 3. 2) but originally due to Plotkin ( Plo80] Proposition 1, see also [Sta85] Fundamental Theorem of Logical Relations) Theorem 2.4.4 (The Binary Logical Relations Theorem) Let I and J be interpretations and let R : I J be a logical relation. Suppose that c I R c J for each and for each c : Then for all oe, for all e : oe, ....

....lattice interpretation of a type corresponds to a per over the standard domain interpretation. To do this we have to use a more general notion of logical relation. 3.3 Ternary Logical Relations The Binary Logical Relations Theorem (2.4. 4) is actually an instance of a more general theorem ( Plo80] which concerns relations between interpretations, where is any ordinal. Although the general theorem is straightforward to state and prove, it is rather cumbersome notationally. The only other instance which we need to consider is the ternary case, which we now describe. Definition 3.3.1 ....

G. Plotkin. Lambda-definability in the full type hierarchy. In J.P. Seldin and J.R. Hindley, editors, To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus, and Formalism, pages 363--373. Academic Press, New York, 1980.

....all parameter values to a finite check. One of the most important methods for proving data independence theorems is setting up relations or mappings between the behaviours that two different versions of a parameterised system can display. This is an application of the theory of logical relations [15, 17, 18, 23]. Specifically, we can ask the question of when, if T and T 0 are two values for a type parameter, and OE : T T 0 is a function, does the function OE lift to map each behaviour of the parameterised process P(T ) to one of P(T 0 ) Given that we are concentrating on traces, this can be ....

G.D. Plotkin, Lambda-definability in the full type hierarchy, in `To H.B. Curry: essays on combinatory logic, lambda calculus and formalism' (Seldin and Hindley, eds), Academic Press, 1980.

....all parameter values to a finite check. One of the most important methods for proving data independence theorems is setting up relations or mappings between the behaviours that two different versions of a parameterised system can display. This is an application of the theory of logical relations [22, 24, 25, 34]. Specifically, we can ask the question of when, if T and T 0 are two values for a type parameter, and OE : T T 0 is a function, does the function OE lift to map each behaviour of the parameterised process P (T ) to one of P (T 0 ) Given that we are concentrating on traces, this can be ....

G.D. Plotkin, Lambda-definability in the full type hierarchy, in `To H.B. Curry: essays on combinatory logic, lambda calculus and formalism' (Seldin and Hindley, eds), Academic Press, 1980.

....V nat op the vertical natural numbers flipped upside down. The V nat component was concerned exclusively with a form of snapback. It is instructive to examine the parametricity argument giving the final isomorphism with N . It is in fact the usual parametricity argument for Church numerals [Plotkin 1980; Reynolds 1983] extended to take appropriate care of the presence of . Let p 2 [ 8fi: fi Gammaffi fi) fi Gammaffi fi) and consider any flat cpo D, d 2 D, and c : D Gammaffi D. Let R : N D be the relation consisting of pairs h ; i and hm; c m (d)i. Then hsucc; ci 2 R Gammaffi R ....

Plotkin, G. D. 1980. Lambda-definability in the full type hierarchy. In J. P. Seldin and J. R. Hindley Eds., To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, pp. 363--373. Academic Press.

....of the tree Without a theoretical understanding of the notion of polytypism it is difficult to provide convincing arguments for one or the other choice. This paper contributes to the theoretical foundations of polytypism, albeit tentatively. We draw inspiration from Reynolds [29] and Plotkin s [28] seminal accounts of the semantics of parametric polymorphism. Roughly speaking, Reynolds and Plotkin showed that any parametrically polymorphic function satisfies a certain (di)naturality property that is derivable from the type of the function via so called logical relations . We turn this ....

Gordon D. Plotkin. Lambda-definability in the full type hierarchy. In J.P. Seldin and J.R. Hindley, editors, To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, London, 1980.

....the theory. So we somehow need to add constants to resolve terms, while ensuring extensionality as we go. The construction will occupy the coming sections. As a consequence of our construction we are led to work with what should be considered a syntactic version of Kripke style logical relations ([Plo80, MM91]) The similarity with techniques from intuitionistic semantics is striking. 5 The Theories ABC and BCT As discussed in the previous section, the axioms of ABC will not imply that all elements inhabiting a sum type must in fact be of the form in i x. For example, the (open) term model for the ....

G. D. Plotkin. Lambda definability in the full type hierarchy, in: P. Seldin and R. Hindley, eds., To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pp. 363-373. Academic Press, New York, 1980.

....products and finite sums. Our techniques are based on concepts from topos theory, however our exposition is elementary. Introduction The use of logical relations as a tool for characterising the definable elements in a model of the simply typed calculus originated in the work of Plotkin [10], who obtained such a characterisation of the definable elements in the full type hierarchy using a notion of Kripke logical relation. Subsequently, the more general notion of a Kripke logical relation of varying arity was developed by Jung and Tiuryn, and shown to characterise the definable ....

G.D. Plotkin. Lambda-definability in the full type hierarchy. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, New York, 1980.

.... 17, 18] The definition of Standard ML constitutes an extensive experiment in programming language specification based on operational semantics [46, 45] The method of logical relations is fundamental to the study of the typed calculus, as emphasized by Friedman [23] Statman [64] and Plotkin [57]. Examples of the use of logical relations in the analysis of programming languages can be found in Plotkin s influential study of PCF [56] and in Mitchell s chapter mentioned above, to name two sources. The metaphor of computations (versus values ) implicit in Plotkin s v calculus [55] was ....

Gordon Plotkin. Lambda-definability in the full type hierarchy. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, pages 363--373. Academic Press, 1980.

....relation T (R) T (D) T (E) and (p[D] p[E] 2 T (R) We may regard a relation R : D E as relating different representations of ff, and T (R) as an invariant relationship that must be maintained. Typically, the relation T (R) is determined in an inductive manner (of logical relations [26]) with the significant caveat that free type variables other than ff are mapped to identity relations. The idea is that two pieces of code satisfying invariant T (R) should behave equivalently from the point of view of the visible types, types other than ff. Relational parametricity gives ....

G. D. Plotkin. Lambda-definability in the full type hierarchy. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays in Combinatory Logic, Lambda Calculus and Formalism, pages 363-- 373. Academic Press, 1980.

....important tool used in proving some deep results about various typed calculi and their models. A special form of the concept of a logical relation first appeared in Harvey Friedman s seminal paper [4] General logical relations were defined and used extensively in the pioneering work of Plotkin [18] and Statman [19, 21, 20] and later on in a more general setting by Breazu Tannen and Coquand [2] Mitchell [15] Mitchell and Moggi [16] and Abramsky [1] among others. As the name indicates, logical relations are certain kinds of relations, and they are used to prove relational properties of ....

G.D. Plotkin. Lambda definability in the full type hierarchy. In J. P. Seldin and J. R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 363--373, London, 1980. Academic Press.

....because they see a close relationship between information hiding through type variables (in a polymorphic language) and information hiding through local variables. Our own view is somewhat more technical: We know that logical relations can often be used to characterize the definable functions [27, 6] or if the model consists of dcpo s the limits of definable functions 3 We use the ML notation x for explicitly dereferencing a variable x. 32] Hence we try to use them a priori (as in [20] to construct a model in which all elements are limits of definable ones, so that we obtain ....

....formulation, because our current definition of the sets Sigma L is very technical, but we will come back to this in a moment. 3) means that the family (R ) 2Type is a logical relation [18] for every R 2 Sigma. Logical relations are known to be a useful tool for reasoning about terms [27, 32, 34, 6, 20]. We finally define the support of an element d 2 [ to be the set supp (d) T fL fi fi d 2 [ L g One may wonder whether d 2 [ supp(d) i.e. whether there is a smallest set L with d 2 [ L . We have not examined this question, as it is irrelevant for our purposes. As ....

Gordon D. Plotkin. Lambda-definability in the full type hierarchy. In J.P. Seldin and J.R. Hindley, editors, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 363--374. Academic Press, 1980.

....notations used in the paper. The reader not familiar with simply typed calculus should consult Hindley and Seldin [3] Section 2 presents standard models for simply typed calculus, it is based on Henkin [2] Section 3 presents the Completeness theorem, it is based on Friedman [1] Plotkin [5] [6] and Statman [9] Section 4 presents the construction of a model for some equational theories. Section 5 presents Statman s finite completeness theorem. Both section 4 and 5 are based on [8] Section 6 presents the definability conjecture. The notion of definability is taken from Plotkin [5] ....

....[6] and Statman [9] Section 4 presents the construction of a model for some equational theories. Section 5 presents Statman s finite completeness theorem. Both section 4 and 5 are based on [8] Section 6 presents the definability conjecture. The notion of definability is taken from Plotkin [5] [6]. This conjecture has been studied by Plotkin and Statman. At last section 7 presents the higher order matching conjecture and the proof that the definability conjecture implies the higher order matching conjecture. The decidability of higher order matching is conjectured in Huet [4] the ....

G. D. Plotkin, Lambda-Definability in the Full Type Hierarchy, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J.R. Hindley and J.P. Seldin (Eds.), Academic Press, 1980, pp. 365-373.

....the pitfalls illustrated by Example 1.2. Some background The methods presented here for reasoning about recursive functions and local storage are rooted in the work of O Hearn and Tennent (1995) and Seiber (1995) These authors use relational parametricity (Reynolds 1983) and logical relations (Plotkin, 1973, 1980) to give denotational models of Algol like languages which match the operational behaviour of local variables better than previous models did. Since our goal is not to produce fully abstract models, but rather to identify practically useful proof methods for contextual equivalence, there is some ....

....0 (r 0 ) 4.6) We call this family of relations logical simply because it relates function values if, roughly speaking, they map related arguments to related results. This is the characteristic feature of a wide range of relations used in connection with the lambda calculus which ever since (Plotkin 1973, 1980) have been called logical relations . Note. It is possible to simplify Definition 4.2 by replacing the use of an arbitrary extension r 0 B r by r itself in the defining clauses for E oe (r) and K oe (r) but not V oe oe 0 (r) This simplification depends partly upon the flat nature of ....

Plotkin, G. D. (1980). Lambda-definability in the full type hierarchy. In J. P.

....that a type indexed family of relations is defined by induction on types and a proof like that of the Basic Lemma is part of the construction, but the family of relations defined is not logical. Examples can be found in Plotkin s and Jung and Tiuryn s lambda definability results using I relations [Plo80] and Kripke logical relations with varying arity [JT93] respectively, and Gandy s proof of strong normalization using hereditarily strict monotonic functionals [Gan80] In each of these cases, the family of relations involved turns out to be a pre logical relation (Example 3.12, Sect. 6 and ....

....A 1 or alternatively, for the case of projection and filtering, to fha 1 ; a n ; a 1 ; a 1 i j ha 1 ; a n i 2 Sg. If R is a logical relation then so is 8R, but the projection and filtering of R are not logical relations in general. 2 Example 3.12. Plotkin s I relations [Plo80] give rise to pre logical relations. The family of relations on the full type hierarchy consisting of the tuples which are in a given I relation at a given world (alternatively, at all worlds) is a pre logical relation which is not in general a logical relation. 2 A related example concerning ....

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G. Plotkin. Lambda-definability in the full type hierarchy. In: To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, 363--373. Academic Press (1980).

....that a type indexed family of relations is defined by induction on types and a proof like that of the Basic Lemma is part of the construction, but the family of relations defined is not logical. Examples can be found in Plotkin s and Jung and Tiuryn s lambda definability results using I relations [Plo80] and Kripke logical relations with varying arity [JT93] respectively, and Gandy s proof of strong normalization using hereditarily strict monotonic functionals [Gan80] In each of these cases, the family of relations involved turns out to be a prelogical relation (Examples 3.8 and 3.9) which ....

....composition of any two pre logical relations is pre logical. Then observe that the above relation is just the composition of closed term interpretation in B (which is pre logical according to Example 3.6) and the inverse of closed term interpretation in A. 2 Example 3.8. Plotkin s I relations [Plo80] give rise to pre logical relations. The family of relations on the full type hierarchy consisting of the tuples which are in a given I relation at a given world (alternatively, at all worlds) is a prelogical relation which is not in general a logical relation. Similarly for Jung and Tiuryn s ....

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G. Plotkin. Lambda-definability in the full type hierarchy. In: To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, 363--373. Academic Press (1980).

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G.D. Plotkin. Lambda definability in the full type hierarchy. In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 363--373. Academic Press, 1980.

.... the internal hom of Set, into itself corresponding to no intuitionistic proof of (A A) A A) Strengthening dinaturality to logicality suffices to eliminate these spurious transformations, leading to a full completeness result for the intuitionistic categorical logic of the category of sets [Plo80]. In this paper we expose a highly parallel situation for the linear categorical logic of the category of Chu spaces over 2. Elsewhere we have shown [Pra98] that, while ordinary dinatural transformations are strong enough for full completeness of the fragment of MLL in which each variable in the ....

G.D. Plotkin. Lambda definability in the full type hierarchy. In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 363--373. Academic Press, 1980.

....value assignments of the parameters of the theorem, possibly subject to given axioms. More recently analogous semantic notions of abstract constructive proof have begun to appear, in particular natural and dinatural transformations [LS86, BS96] and related notions such as logical transformations [Plo80], game strategies [AJ94, HO93] and uniformity conditions [Loa94] The naturality condition expresses transformational invariance for all transformations of the parameters of the proof, again possibly subject to given axioms. The interpretation of natural transformations as constructive proofs ....

....one that corresponded to no linear logic proof. The contribution of this paper is that strengthening dinaturality to binary logicality eliminates all spurious transformations. The same strengthening has previously proved successful in eliminating spurious transformations in the category of Sets [Plo80, PR98]. This correspondence between linear logic and Chu spaces enhances both subjects. On the one hand the correspondence furnishes Chu spaces with the attractive and well studied structure of linear logic. On the other, linear logic benefits from having a model that is of interest both in its own ....

G.D. Plotkin. Lambda definability in the full type hierarchy. In To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, pages 363--373. Academic Press, 1980.

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G. D. Plotkin. Lambda-definability in the full type hierarchy. In To H.B.Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism. Academic Press, 1980.

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G.D. Plotkin. Lambda definability in the full type hierarchy. In R. Hindley and J. Seldin, editors, To H.B. Curry: essays in Combinatory Logic, lambda calculus and Formalisms. Academic Press, 1980.

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G. D. Plotkin, Lambda-Definability in the Full Type Hierarchy, To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, J.R. Hindley and J.P. Seldin (Eds.), Academic Press, 1980, pp. 365-373.

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G.D. Plotkin, "Lambda-Definability in the Full Type Hierarchy", in J.P. Seldin, J.R. Hindley eds., To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, London 1980, 363-373.

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G. Plotkin. Lambda-definability in the full type hierarchy. In: To H.B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism, 363--373. Academic Press (1980).

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